This question involves a circle equation and a line, and requires solving both parts a) and b).

a) Determining the radius and coordinates of the center of the circle:

The equation given is:
$x^2 - 6x + y^2 + 2y + 1 = 0$ This is not yet in standard form for a circle, which is typically expressed as:
$(x - h)^2 + (y - k)^2 = r^2$ Where $(h, k)$ is the center of the circle and $r$ is the radius.

We need to complete the square for both $x$ and $y$ terms.

b) Investigating if the line $3x - 4y + 3 = 0$ is a tangent to the circle:

A line is tangent to the circle if the perpendicular distance from the center of the circle to the line is equal to the radius. This can be calculated using the formula for the distance from a point $(x_1, y_1)$ to a line $Ax + By + C = 0$:

$d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$

Now I will start solving the equation. sympy import symbols, Eq, sqrt

Define variables

x, y = symbols('x y')

The equation of the circle

circle_eq = Eq(x2 - 6*x + y2 + 2*y + 1, 0)

Completing the square for x and y terms

x^2 - 6x becomes (x - 3)^2 - 9

y^2 + 2y becomes (y + 1)^2 - 1

Rewrite the equation of the circle

center_x = 3 center_y = -1 radius_squared = 9 + 1 - 1 # Calculated after completing the square

Now, calculate the radius

radius = sqrt(radius_squared)

The equation of the line: 3x - 4y + 3 = 0

A, B, C = 3, -4, 3 # Coefficients of the line equation distance_to_line = abs(A * center_x + B * center_y + C) / sqrt(A2 + B2)

center_and_radius = (center_x, center_y, radius, distance_to_line) center_and_radius### a) The radius and the center of the circle: After completing the square, the equation of the circle becomes: $(x - 3)^2 + (y + 1)^2 = 9$ Thus, the center of the circle is $(3, -1)$, and the radius is $3$.

b) Investigating if the line $3x - 4y + 3 = 0$ is a tangent to the circle:

The perpendicular distance from the center $(3, -1)$ to the line $3x - 4y + 3 = 0$ is $\frac{16}{5}$, which is approximately 3.2.

Since this distance (3.2) is greater than the radius (3), the line is not a tangent to the circle.

Would you like to go over any of the steps in more detail or have further questions on this?

Here are some related questions for you to explore further:

1. How do we complete the square for both $x$ and $y$ terms in general?
2. What are some geometric interpretations of a circle’s center and radius in coordinate geometry?
3. How can you find the intersection points of a line and a circle, if they exist?
4. What other conditions might affect whether a line is tangent to a circle?
5. Can you describe how a point on the circle relates to its equation?

Tip: Always check whether the distance from the center to a line is equal to the radius when determining if the line is tangent.